The set $P$ consists of complex trigonometric polynomials $$f_k(t)=\sum_{j=0}^k a_j e^{iτ_jt},$$ where $$ k\in \mathbb N, a_1,...,a_k \in \mathbb C, τ_1,...,τ_k \in \mathbb R.$$
Show that $P$ is a complex linear space.
I have to show that $$f,g \in P, α, β \in \mathbb C \Rightarrow (αf+βg) \in P.$$
My solution: $$(αf+βg)_k(t) = \sum_{j=0}^k (αa_j e^{iτ_jt} + βa_j e^{iτ_jt} ) = \sum_{j=0}^k αa_j e^{iτ_jt} + \sum_{j=0}^k βa_j e^{iτ_jt} \\ = α\sum_{j=0}^k a_j e^{iτ_jt} + β\sum_{j=0}^k a_j e^{iτ_jt} = αf_k(t) + βg_k(t). $$ And it is true.
Question: Is my solution correct?